On the braided Fourier transform in the n-dimensional quantum space

We work out in detail a theory of integrability on the braided covector Hopf algebra and the braided vector Hopf algebra of type A(n) introduced by Majid. Using a braided Fourier transform very similar to the one defined by Kempf and Majid we obtain n-dimensional analogs of results by Koornwinder expressing the correspondence between products of the q^2-Gaussian g_{q^2}((x)) times monomials, and products of the q^2-Gaussian G_{q^2}(\partial) times q(2)-Hermite polynomials under the transform. We invert the correspondence by finding a suitable inversion, different from the one of Kempf and Majid. We show that with this transforms, whenever n greater than or equal to 2, the Plancherel measure will depend on the parity of the power series that we are transforming.